A study of propagation of cosh-squared-Gaussian beam through ...

Optica Applicata, Vol. XLI, No. 4, 2011
A study of propagation of cosh-squared-Gaussian
beam through fractional Fourier transform systems
SOMAYE SADAT HASHEMI 1* , SAEED GHAVAMI SABOURI 2 , MAHMOOD SOLTANOLKOTABI 3
1 Falavarjan Branch, Islamic Azad University, Isfahan, Iran
2 Department of Physics, University of Isfahan, Isfahan, Iran
3 Quantum Optics Group, Department of Physics, University of Isfahan, Isfahan, Iran
* Corresponding author: hashemi.somaye@gmail.com
In this paper, we consider properties of cosh-squared-Gaussian beam passing through ideal and
apertured fractional Fourier transforms (FRFT) systems. We use Collins integral formula and
the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian
functions. These studies indicate that the normalized intensity distributions with FRFT order are
periodic. The variation period is 2 and is independent of the impact of aperture.
Keywords: cosh-squared-Gaussian, fractional Fourier transforms (FRFT).
1. Introduction
The Fourier transform is one of the most important mathematical tools that are used
in physical optics, optical information processing, linear system theory, and some
other areas.
The fractional Fourier transform (FRFT) is regarded as a generalization of
the conventional Fourier transform. There were some reports about the relation between
FRFT and quantum mechanics in the 1980s [1, 2], but those did not gain much
attention. MENDLOVIC and OZAKTAS had introduced FRFT into optics [3, 4], then
LOHMANN studied the relation between the image rotation in the optical system and
the FRFT [5]. He introduced two optical setups for performing a fractional Fourier
transform.
The application of the FRFT in other areas such as signal processing, beam shaping
and image encryption, has gained more attention [6–11]. Moreover, the propagations
of laser beams through FRFT systems have been widely investigated [12–21].

898 SOMAYE SADAT HASHEMI et al.
The optical field distributions on FRFT plane can be derived from Collins integral
formula [22]. Although the lenses in FRFT systems are infinite, one can use in real
life apertured lenses or other limited optical instruments. Thus, it is very important
and necessary to consider the hard aperture FRFT systems [23–26].
There are some reports about beams passing through apertured FRFT systems
[27–32], and also the effect of FRFT order and aperture size on the intensity distribution
for two optical setups of Lohmann [27–32]. In these reports, they did not point
out the variation period of intensity with FRFT order. However, CHEN et al. have shown
the effect of the variation period of intensity on the FRFT order [33]. These authors
have used the type I Lohmann system to achieve the FRFT of cosh-squared-Gaussian
(CSG) beam, and they have studied the properties of CSG beam passing through the
ideal and aperture FRFT systems. The characteristics of cosh-Gaussian beam have also
been widely studied [21, 34–36].
In this paper, we have investigated the properties of CSG beam passing through
the ideal and aperture type II Lohmann system, and we have studied the intensity
distribution of CSG beam on FRFT plan. To do this, we have used two different
methods, analytical formula and Collins diffraction integral formula. We have implemented
these methods for both Lohmann systems (types I and II) and then have
compared the results.
The paper is organized as follows: the theoretical analyses of CSG beam passing
through ideal and apertured FRFT systems are given in Section 2. The numerical
comparisons using the analytical formulae and the diffraction integral formulae are
given in Section 3. Finally, our conclusion is given in Section 4.
2. Field distribution calculation for ideal and analytical cases
Let us consider the case of an infinite size of lenses in FRFT systems and Lohmann
systems as illustrated in Fig. 1. We have used type II Lohmann system, as shown in
Fig. 1b, where f s is the standard focal length, p is the FRFT order, φ = pπ/2, d is
the distance between the input (z = 0) and output (z = d) planes.
Input
plane
d = f s tan(φ/2)
f = f s /sinφ
d = f s tan(φ/2)
Output
plane
Fig. 1. Lohmann optical systems: type I (a), and type II (b).
a f = fs/tan(φ/2) b
d = f s sinφ
z = 0 z = d

A study of propagation of cosh-squared-Gaussian beam... 899
I/I max
1.0
0.8
0.6
0.4
0.2
0.0 –10 –5 0 5 10
It is known that the optical field distribution of the one-dimensional cosh-squared-
-Gaussian beams on the input plane is characterized by [33, 36]:
E0( x0) =
x 0 [mm]
2
⎛ x0 ⎞ 2
exp⎜–
----------- ⎟cosh
( Ω x0)
2
⎝ w ⎠
0
Ω = 0
Ω = 0.5
Ω = 0.7
Ω = 0.9
Ω = 1.0
Fig. 2. Normalized intensity for different Ω ’s on input plane (w 0 = 2.5 mm).
where w 0 represents the beam waist of Gaussian beam, Ω is the parameter associated
with the cosh part, and x 0 is the transversal position on the plane z = 0. By changing
the value of Ω, one gets different optical field distributions (Fig. 2). For Ω =0, Eq.(1)
denotes the usual Gaussian beam.
To achieve the optical field distributions on FRFT plane we have use the Collins
integral formula [22]:
Ex ( )
=
∞
i
– ------------ E0( x0) i
λB
π
------------ ⎛ 2 2
Ax0 + Dx – 2xx ⎞
∫ exp – 0 dx
λB ⎝ ⎠ 0
– ∞
The constant phase in Collins formula, which has no influence on the output
intensity distribution has been omitted. A, B, and D are the elements of the system
transfer matrix.
If the lenses of type II Lohmann system are infinite, the transfer matrix from
the plane at z =0 to plane z = d becomes:
d
1 – --------- d
f
M
d
f 2
= =
2
------------ – --------d
1 – --------f
f
cosφ fs sinφ
1
– -------- sinφ
cosφ
f s
(1)
(2)
(3)

900 SOMAYE SADAT HASHEMI et al.
By substituting relations (3) and (1) into relation (2), and performing the integration
we obtain the optical field distribution on FRFT plane
Eideal( x)
=
2
iπw 0
– ----------------------------------------------------------------
2
4( λ fssinφ + iπw 0 cosφ
)
( πxw0) 2
iπx
– -----------------------------------------------------------------------------------
2
λ fs sinφλf
( ssinφ + iπw 0 cosφ
)
2
cosφ
× exp
– ---------------------------λ
fs sinφ
Ω
1
(4)
From relation (4) we see that the optical field distribution on FRFT plane in addition
to the beam parameters depends on the system parameters such as the standard focal
length fs and FRFT order p.
It is worth mentioning that the transfer matrixes for types I and II Lohmann systems,
in the case of an infinite size of lenses, are equal. In other words, the ideal optical field
distribution for both Lohmann systems are identical. In this case, we have obtained
the same distribution of the ideal field on FRFT plane for both Lohmann systems. Our
results are for type I Lohmann system the same as given in Ref. [33].
Usually, the lens in FRFT system is finite and an aperture is to be added in
the calculation. We consider two apertures, one in front of the input lens and the other
on the output lens. The second aperture only truncates the output field distribution.
According to the Collins diffraction integral formula the approximate analytical
expression for the output field distribution of a CSG in the FRFT plane is derived as
follows:
2 ⎛ 2
w0 λ fs sinφ
⎞ ⎛ 2
⎜-------------------------------------------------------- ⎟
2πiw0Ω x ⎞
× + exp
cosh⎜--------------------------------------------------------
⎟
⎜ 2 ⎟ ⎜ 2 ⎟
⎝ λ fssinφ + iπw 0 cosφ
⎠ ⎝ λ fssinφ + iπw 0 cosφ
⎠
Eapertured( x)
=
where a denotes the half-width of the lens aperture, defined by the hard aperture
function as:
tx ( )
=
⎧ 1, x ≤ a
⎨
⎩ 0, x > a
In this case, the relation (5) becomes
Eapertured( x)
=
a
×
i
– ------------ E0( x0) i
λB
π
------------ ⎛ 2 2
Ax0 + Dx – 2xx ⎞
∫ exp – 0 dx
λB ⎝ ⎠ 0
– a
∞
i
– ------------ tx ( 0)E(
x
λB
0 0)
i
(7)
π
------------ ⎛ 2 2
Ax0 + Dx – 2xx ⎞
∫
exp – 0 dx
λB ⎝ ⎠ 0
– ∞
×
(5)
(6)

A study of propagation of cosh-squared-Gaussian beam... 901
In order to calculate the integral we should select a form of hard aperture function
introduced in many references [23–26]. We use one given in Ref. [24]:
tx ( ) An (8)
a
where An and Bn are the expansion and Gaussian coefficients, respectively, which can
be obtained directly from computation of relations [25, 26].
By substituting relations (1), (3), and (8) into relation (7), and performing tedious
integration:
2
----------- x 2
N ⎛ ⎞
= ∑ exp⎜–
⎟
n = 1 ⎝ ⎠
x
(9)
an approximate analytical expression for the output field distribution in the FRFT plane
is obtained:
2n
α x 2
∞
⎛– – 2βx⎞ ∫ exp
dx
⎝ ⎠
– ∞
⎛ β
– --------- ⎞
⎝ α ⎠
2n π β-------α
2
⎛ ⎞ ( 2n)!
α
⎜---------- ⎟ ---------------------------------
⎝ α ⎠ l! ( 2n– 2l)!
4β 2
⎛ ⎞
⎜-------------- ⎟
⎝ ⎠
l
=
n
exp ∑
l = 0
where
Ex ( ) = An n = 1
×
N
∑
B n
2
iπw 0 ( πxw0) – ----------------
4ξn 2
iπx
– ------------------------------λ
fsξ n sinφ
2
cosφ
exp
– --------------------------λ
fs sinφ
Ω
1
2 2 2
⎛ w0 λ fs sinφ
⎞ ⎛ 2πiw0Ω x ⎞
+ exp⎜------------------------------------------
⎟cosh⎜------------------------------
⎟
⎝ ξn ⎠ ⎝ ξn ⎠
2
2 Bnw 0λ
fs sinφ
ξn λ fs sinφ iπw 0 cosφ
a 2
= + + ----------------------------------------
a
δ =
----------w0
Relations (10) to (12) are the general expressions, which are valid within
the paraxial approximation. Then, apart from the standard focal length f s and
FRFT order p, the intensity distributions on FRFT plane depend on the truncation
parameter δ as well.
We see that whenever a/w 0 →∞, Eq. (10) reduces to Eq. (4). Also, for Ω =0
Eqs. (4) and (10) reduce to the optical field distributions of Gaussian beam passing
through ideal and apertured FRFT systems, respectively, as one expects.
Although Eqs. (10) to (12) are the approximate analytical expressions, they provide
a more convenient method for studying the propagation characteristics of a flattened
Gaussian beam through the two types of apertured FRFT systems than those using
the diffraction integral formula directly.
×
(10)
(11)
(12)

902 SOMAYE SADAT HASHEMI et al.
3. Numerical and analytical analyses
In order to compare our result given by expression (10) with those given by
the diffraction integral formula (5), we have performed numerical calculations. We
have particularly paid attention to the truncation parameter and FRFT order of the normalized
intensity distributions. We have also compared our result with that of type I
Lohmann system [33]. In these numerical calculations, we have used λ =1.06μm,
f s =1000mm, w 0 =2.5mm, Ω = 0.6, N =10, δ =0.7, with A n and B n being taken from
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
p = 0.1
–2 –1 0 1 2
x [mm]
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
p = 0.5
–1 0 1
x [mm]
p = 0.9
–1 0 1
x [mm]
Fig. 3. Normalized intensity distributions of CSG beam on FRFT plane. The dotted lines represent
the case of using the formula (10), and the solid lines denote the case of using the diffraction integral
formula (5) for different FRFT order values.
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
–1
p = 0.3
0
x [mm]
1
p = 0.7
–1 0 1
x [mm]
p = 1.0
–1 0 1
x [mm]

A study of propagation of cosh-squared-Gaussian beam... 903
I/I max
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
p = 0.3, δ = 1
–2 –1 0
x [mm]
1 2
p = 0.5, δ = 1
–2 –1 0 1 2
x [mm]
p = 0.7, δ = 1
–2 –1 0
x [mm]
1 2
0
–1
p = 0.9, δ = 1
0
x [mm]
1
I/I max
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
Fig. 4. The same as Fig. 3,but for δ = 0.5 and δ =1.
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
p = 0.3, δ = 0.5
–1 –0.5 0 0.5 1
x [mm]
p = 0.5, δ = 0.5
–1 –0.5 0 0.5 1
x [mm]
p = 0.7, δ = 0.5
–1 –0.5 0 0.5 1
x [mm]
p = 0.9
δ = 0.5
–1 –0.5 0 0.5 1
x [mm]

904 SOMAYE SADAT HASHEMI et al.
Ref. [25, 26]. Figure 3 shows the normalized intensity distributions of CSG beam on
FRFT plane.
From these figures, we reach a very good agreement between two approaches, i.e.,
numerical and analytical calculations, especially for p > 0.9. It is worth mentioning
that numerical calculation of approximate analytical formula was much faster than
the numerical integral calculation.
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
p = 0.5
p = 0.9
p = 0.1
Type I
Type II
–1 0 1
x [mm]
–1 0 1
x [mm]
0.0
–1 0
x [mm]
1
I/I max
I/I max
I/I max
p = 0.3
–1 0 1
x [mm]
Fig. 5. Normalized intensity distributions of CSG beam on FRFT plane by using the analytical formula
for type I and II Lohmann systems.
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
p = 0.7
–1 0 1
x [mm]
p = 1.0
0.0
–1 0
x [mm]
1

A study of propagation of cosh-squared-Gaussian beam... 905
Next, we have examined the effect of δ on the coincidence of diffraction integral
and analytical simulation. We have compared the results for δ =0.5 and δ =1 and
p = 0.3, 0.5, 0.7 and 0.9 (Fig. 4).
We see that for the same value of p and for smaller values of δ the coincidence of
figures is very good. But, for higher values of δ (and fixed p) the agreement is
poor.
Next, we consider the variations of normalized intensity distributions of CSG beam
on FRFT plane by using the analytical formula for type I and II Lohmann systems, by
considering that the two transfer matrixes of these systems are equal in an ideal case.
We can seen from Fig. 5, when p is near 1 (p ≥ 0.7), that the simulation results for both
Lohmann systems show good coincidence, especially on the Fourier transform (p =1)
plane.
It is worth mentioning that similar result for flattened Gaussian beams was
reported [27]. The variations of normalized intensity distributions of CSG beam on
FRFT plane by using the analytical formula for type I and II Lohmann systems, for
p = 1 and δ = 0.5, 0.7 and 1 are shown in Fig. 6.
I/I max
1.0
0.8
0.6
0.4
0.2
δ = 0.5
δ = 1.0
p = 1.0
Type I
Type II
δ = 0.7
0.0
–1 0
x [mm]
1
Fig. 6. Normalized intensity distributions of CSG beam on FRFT plane by using the analytical formula
for type I and II Lohmann systems.
Now, we consider the variations of normalized intensity distributions of CSG beam
for various p and δ, and compare them with ideal case. Because the intensity
distributions are symmetric about the vertical-axis, so we have only drawn the normalized
intensity distribution on the right of vertical axis in the Fig. 7.
In this figure, we find for δ > 3, that the normalized intensity distributions are
similar to the ideal case and this is especially seen for δ ≥ 8. We found that the variation
period of the normalized intensity distributions versus p, is 2 for all the values of δ.
In this system, the variation period of the normalized intensity distributions does not
depend on δ, however, in type I Lohmann system the variation period of the normalized
intensity distributions does depends on δ [33].

906 SOMAYE SADAT HASHEMI et al.
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
0.0
0 1
1.0
0.8
0.6
0.4
0.2
0.0
0
1.0
0.8
0.6
0.4
0.2
0.0
0 1
Fig. 7. Variations of normalized intensity distributions with FRFT order after a CSG beam passing
through different FRFT systems.
Furthermore, the variations of normalized intensity distributions with Ω after
the CSG beams passing through different FRFT systems are presented in Fig. 8. It is
shown that the normalized intensity distributions on FRFT plane strongly depend on
the initial beam parameters Ω in addition to FRFT order p and δ.
4. Conclusions
2
2
2
Ideal
δ = 0.7
δ = 1.0
δ = 2.0
δ = 3.0
δ = 8.0
p = 0.45
3 4 5 6
x [mm]
3
4
p = 2.45
x [mm]
p = 3.45
6
4 5
x [mm]
Based on the Collins integral formula and the fact that a hard aperture function can be
expanded into a finite sum of complex Gaussian functions, the propagation properties
I/I max
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
0.0
0 1
1.0
0.8
0.6
0.4
0.2
0.0
0 1
1.0
0.8
0.6
0.4
0.2
0.0
0 2
4
2
2
x [mm]
x [mm]
p = 0.75
x [mm]
p = 2.75
3
p = 3.75
6
3

A study of propagation of cosh-squared-Gaussian beam... 907
I/I max
I/I max
1.0
0.8
0.6
0.4
0.2
0.0
0 0.5
1.0
0.8
0.6
0.4
0.2
p = 0.45, δ = 0.7
1.0
x [mm]
p = 1.0, δ = 0.7
0.0
0 0.5 1.0
x [mm]
Ω =
0
0.5
0.9
Fig. 8. Variations of normalized intensity distributions as a function of Ω.
1.5
1.5
I/I max
I/I max
Ω =
0
0.5
0.9
of CSG beam passing through ideal and apertured type II Lohmann system on FRFT
plane, have been studied and simulated.
By comparing the results obtained by analytical formula and diffraction integral
we have reached the following conclusions. First, when p is near to one and any other
odd number, the normalized intensity distributions obtained by using the approximate
analytical and the numerical integral formulas coincide exactly. Moreover, when p is
near to 1 (p ≥ 0.7), the simulation results for type I and II Lohmann systems highly
coincide, especially on the Fourier transform (p = 1) plane.
Second, for δ < 3, the intensity distribution is very match dependent on the value
of δ but for δ ≥ 8, this dependence is removed. Also, from Fig. 4 we see that
the smaller δ causes the better match of the two different methods (analytical and
the numerical integral formulas). Contrary to what has been reported in Ref. [33],
when δ < 6,the aperture has a great impact on the normalized intensity distributions,
and the variation period of normalized intensity distributions with FRFT order is 4; on
the other hand, when δ > 6, the impact of aperture can be ignored, and the variation
period of normalized intensity distributions with FRFT order is 2, as shown in Fig. 7.
We have not seen any dependency of intensity distributions of the FRFT plane on
the value of δ, which may be due to the existence of two apertures instead of one. Also,
Figure 8 shows that the normalized intensity distributions on FRFT plane strongly
depend on the initial beam parameter Ω.
1.0
0.8
0.6
0.4
0.2
0.0
0 2
1.0
0.8
0.6
0.4
0.2
0.0
0 0.05
p = 0.45, δ = 8.0
4 6 8
x [mm]
p = 1.0, δ = 8.0
0.10 0.15 0.20
x [mm]

908 SOMAYE SADAT HASHEMI et al.
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Received April 30, 2011
in revised form June 22, 2011